On the range of Carmichael's universal-exponent function

Florian Luca; Carl Pomerance

Displaying similar documents to “On the range of Carmichael's universal-exponent function”

On sum-product representations in $ℤ_{q}$

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

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The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$ , by performing sum and product set operations starting from a given subset $A$ of $ℤ_{q}$ . We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $| A | \sim log q$ ). Roughly speaking we show that all residue classes are obtained from a $k$ -fold sum of an $r$ -fold product set of $A$ , where $r ≪ log q$ and $log k ≪ log q$ , provided the...

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

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We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

Sums of positive density subsets of the primes

Kaisa Matomäki (2013)

Acta Arithmetica

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We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1 - o (1)) α / (e^{γ} l o g l o g (1 / β))$ , which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ *_{m}$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B \subseteq ℤ *_{m}$ of densities α...

Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

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For any given positive integer k, and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

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For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ for all n ≥ n₀, we have $r_{1, k} (A, n) \to \infty$ as n → ∞.

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz, Konrad Zdanowski (2011)

Fundamenta Mathematicae

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We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .

A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

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Let K be an algebraic number field with non-trivial class group G and $_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_{k} (x)$ denote the number of non-zero principal ideals $a_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_{k} (x)$ behaves for x → ∞ asymptotically like $x {(l o g x)}^{1 - 1 / | G |} {(l o g l o g x)}^{_{k} (G)}$ . We prove, among other results, that $₁ (C_{n ₁} \oplus C_{n ₂}) = n ₁ + n ₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Uniform algebras and analytic multifunctions

Zbigniew Slodkowski (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Dati due elementi $f$ e $g$ in un'algebra uniforme $A$ , sia $G=f(M_{A}/f(\partial_{A})$ . Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda\to\log\max g(f^{-1}(\lambda))$ è subarmonica su $G$ e che l’applicazione $\lambda\to g(f^{-1}(\lambda))$ è analitica nel senso di Oka.

Equilibrium states for interval maps: the potential $- t log | D f |$

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $f : I \to I$ be a $C^{2}$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $φ_{t} : x \mapsto - t log | D f (x) |$ for $t$ close to $1$ , and also that the pressure function $t \mapsto P (φ_{t})$ is analytic on an appropriate interval near $t = 1$ .

Limits of log canonical thresholds

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $𝒯_{n}$ denote the set of log canonical thresholds of pairs $(X, Y)$ , with $X$ a nonsingular variety of dimension $n$ , and $Y$ a nonempty closed subscheme of $X$ . Using non-standard methods, we show that every limit of a decreasing sequence in $𝒯_{n}$ lies in $𝒯_{n - 1}$ , proving in this setting a conjecture of Kollár. We also show that $𝒯_{n}$ is closed in $𝐑$ ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

A generalized Kahane-Khinchin inequality

S. Favorov (1998)

Studia Mathematica

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The inequality $ʃ l o g | \sum a_{n} e^{2 π i φ_{n}} | d φ_{1} \dots d φ_{n} \geq C l o g (\sum | a_{n} {|^{2})}^{1 / 2}$ with an absolute constant C, and similar ones, are extended to the case of $a_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2 π i φ}$ .

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

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An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^{2}$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that $\sum_{n \leq x} d ² (n) = x P (l o g x) + E (x)$ , where P(y) is a cubic polynomial in y and $E (x) = O (x^{3 / 5 + ε})$ , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E (x) = O (x^{1 / 2 + ε})$ . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E (x) = O (x^{1 / 2} (l o g x) ⁵ l o g l o g x)$ . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E (x) = O (x^{1 / 2} (l o g x) ⁵)$ . ...